Listen, listen, listen. Be sure to read the CD liner notes as you listen.

See the class handout on symmetry in music.
A horizontal reflection is known as an inversion in music.

See the class handout on symmetry in music.
A vertical reflection is known as a retrograde in music.

Inverses are not in the group. For example, 4-1 = 1/4 but 1/4 is not
an integer.

If a musical motif has symmetry H (inversion about the middle of the staff) and symmetry V (retrograde),
then it will also have the symmetry obtained by composing these together.
Since H*V=R180, the motif will also be symmetric under a 180 degree rotation.
If the inversion is not about the center of the staff, then the motif will be the same as its retrograde-inversion,
but not identical to a 180 degree rotation.

Both Dn (the dihedral group of degree n) and Sn (the symmetric group on n symbols) are examples of non-commutative groups.

S is not closed under addition because the sum of two odd integers is an even integer. S is closed under multiplication,
however, because the product of two odd integers is still odd. S does not form a group under multiplication because inverses,
which are the reciprocals of the odd integers, are not contained in S. Thus, S fails to satisfy property 4 in the definition of
a group.

The multiplication table for G is shown in Table 5.3 on p. 194 of the course text. The inverses are as follows:
0-1 = 0, 1-1 = 3 since 1 + 3 = 0 (mod 4), 2-1 = 2 since 2 + 2 = 0 (mod 4), and
3-1 = 1 since 3 + 1 = 0 (mod 4). G is identical (isomorphic) to the subgroup of D4 given by the four rotations
{e, R90, R180, R270} (compare with the portion of the multiplication table for D4 in the
upper left block).

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Fn = Fn-1 + Fn-2. The golden ratio is the
ratio that subdivides a segment into two parts so that the ratio of the whole to the larger part is equivalent to the
ratio of the larger part to the smaller part. The exact value of the golden ratio, usually symbolized
by φ, is φ = (1 + sqrt{5})/2. The ratio of
consecutive Fibonacci numbers approaches (limits on) the golden ratio as n gets larger and larger. Fibonacci numbers show
up in nature in the number of opposing spirals on pine cones and sunflowers (typically consecutive Fibonacci numbers).
They also occur as the number of petals in many flowers.

φ2 = (3 + sqrt{5})/2, φ3 - 1 = 1 + sqrt{5}, φ4 - 3 φ = 2.

See the class handout on change ringing (or Section 6.1.2 of the course text).

(1 2 3), (2 1 3), (2 3 1), (3 2 1), (3 1 2), (1 3 2), (1 2 3). There are only two legitimate
extents because at any given stage there are only two permissible moves: a = (1 2) or b = (2 3).
Since a2 = e and b2 = e, we must alternate between a and b between changes. This
results in either (ab)3 or (ba)3 as the only possible extents.

See the class handout on change ringing (or Section 6.1.5 of the course text).
This isn't hard to learn if you recall our special dance class with Liz Lerman from last semester.
The moves used to create Plain Bob Minimus are [(ab)3a c]3.

6! = 720.

88. Recall that the number of permissible moves on n bells is Fn+1 - 1. See Table 6.2 of the course text.